## Abstract

It is known (Berry and Dennis 2007 *J. Phys. A: Math. Theor.* **40** 65–74; Berry and Dennis 2012 *Eur. J. Phys.* **33** 723–731)
that only one kind of reaction between wave vortices can occur
generically in a monochromatic optical field. It appears either in
elliptic form as the birth and death of vortex rings or in hyperbolic
form as reconnection between separate vortex lines. To make it occur the
field must be changed, and, since the codimension is one, it suffices
to adjust a single external parameter. The paper analyses a model in
which the initial field is produced by superposing *n* plane waves
of the same frequency but different random amplitudes, directions and
phases. This is perturbed by an additional plane wave of variable
amplitude. The field necessarily obeys the Helmholtz equation and, in
spite of the randomness, there is systematic behaviour for *n* = 3 and 4, which leads to some understanding of the more complicated results for higher values of *n*.
Three plane waves of equal amplitude, perturbed by a fourth, provide a
surprising special case, and the remarkable succession of events
discovered by (O'Holleran *et al* 2006a *J. Eur. Opt. Soc. Rapid Publ.* **1** 06008; O'Holleran *et al* 2006b *Opt. Express* **14** 3039–3044)
is fully explained. This is a central point of the paper. Looking at
the singularity itself, and initially following Berry and Dennis, the
simplest model that satisfies the Helmholtz equation is presented and
also the most general local model that uses 'polynomial waves'. We also
consider waves that are described simply by a polynomial without any
exponential factor. The inclusion of time in the polynomial allows
explicitly for quasi-monochromatic waves in which the events occur
spontaneously, rather than by adjusting an external control. The
circulating phase structure around a simple wave vortex is its most
distinctive feature. But in reconnection two such singular vortex lines
cross one another and the phase pattern around them must reflect this
higher singularity. How it does so is illustrated in the paper.

Original language | English |
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Article number | 105602 |

Number of pages | 11 |

Journal | Journal of Optics |

Volume | 18 |

Issue number | 10 |

Early online date | 31 Aug 2016 |

DOIs | |

Publication status | Published - Oct 2016 |